Condition number and MIMO systems

Wireless communication system often requires solution of linear algebraic equation like y = H*x + w. Where H is channel matrix, x is transmitted signal, y is received signal and w is noise.

  • If small change in H results in small change in y, we say system of equation is well-conditioned.
  • If small change in H results in large change in y, we say system of equation is ill-conditioned.

It is to note that sometimes we say that Matrix “H” is ill-conditioned or well-conditioned (Instead of system of equation). In this case, we are only referring to the fact that how sensitive matrix is to its inversion.

In the presence of round-off errors, ill-conditioned systems are inherently difficult to handle.In this regard, Channel matrix condition number is an important indicator to tell how sensitive a Matrix or solution of a system of equation is to  round-off errors or input to the system. . If the condition number of a matrix is large, the matrix is said to be ill-conditioned.

Higher order MIMO detector systems require to solve equation with large matrix dimensions. And. things become even worse when these detectors are implemented on finite precision machines. So, It becomes very important to know about condition numbers.

Probably following answer by “Mario Carneiro” explains it best.  [ref: https://math.stackexchange.com/q/261300]

Mathematically, if the condition number is less than , the matrix is invertible. Numerically, there are round-off errors which occur. A high condition number means that the matrix is almost  non-invertible. For a computer, this can be just as bad. But there is no hard bound; the higher the condition number, the greater the error in the calculation. For very high condition number, you may have a number round to 0 and then be inverted, causing an error. This is the same thing that would happen if you tried to invert a truly non-invertible matrix, which is why I say that a high condition number may as well be infinite in some cases.

In order to find out if the matrix is really too  ill-conditioned, you should invert the matrix, and then check that AA1=I, to an acceptable precision. There is simply no hard cap on the condition number, just heuristics, which is why your references differ.

So, below are some of the questions that attached video that I created long ago tries to answer.

  • What is Geometric interpretation of  condition number.
  • How to calculate condition number of a matrix.

~Dheeraj

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